tailieunhanh - báo cáo hóa học:" Research Article Optimal Interior Partial Regularity for Nonlinear Elliptic Systems for the Case 1

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Optimal Interior Partial Regularity for Nonlinear Elliptic Systems for the Case 1 | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 680714 23 pages doi 2010 680714 Research Article Optimal Interior Partial Regularity for Nonlinear Elliptic Systems for the Case 1 m 2 under Natural Growth Condition Shuhong Chen1 2 and Zhong Tan2 1 Department of Information and Mathematics Sciences China Jiliang University Hangzhou Zhejiang 310018 China 2 School of Mathematical Science Xiamen University Xiamen Fujian 361005 China Correspondence should be addressed to Shuhong Chen shiny0320@ Received 16 November 2009 Accepted 18 March 2010 Academic Editor Shusen Ding Copyright 2010 S. Chen and Z. Tan. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. We consider the interior regularity for weak solutions of second-order nonlinear elliptic systems with subquadratic growth under natural growth condition. We obtain a general criterion for a weak solution to be regular in the neighborhood of a given point. In particularly the regularity we obtained is optimal. 1. Introduction In this paper we consider optimal interior partial regularity for the weak solutions of nonlinear elliptic systems with subquadratic growth under natural growth condition of the following type - DaAa x u Du Bfx u Dù i 1 . N in Q a 1 where Q is a bounded domain in Rn u and Bi taking values in RN and Aa - has value in RnN. N 1 u Q RNDu Daui 1 a n 1 i N stand for the div of u and 1 m 2. To define weak solution to one needs to impose certain structural and regularity conditions on Aia and the inhomogeneity Bi as well as to restrict u to a particular class of functions as follows for 1 m 2 E1 Aa x u p are differentiable functions in p and there exists L 0 such that dAa x u p dpp l 1 p 2y ì y x u p e Q X Rn X RnN 2 Journal of Inequalities and Applications E2 Aa is uniformly .